Moving to \(W=4\)  leg ladders and cylinders, a still richer variety of behaviors has been observed [1]}, [2]}, [3]}. In particular the nature of the states for a given value of \(x\)  is found to depend significantly on the value of \(t^{\prime }\) .
Under some circumstances, especially in 4-leg cylinders with \(t^{\prime }\)  small and negative,In DMRG studies, a value of \(t^{\prime } \approx - 0.3t\)  is often taken as a value representative of the band-structure of the hole-doped cuprates.
there appears to be [4]}, [5]}, [6]} a Luther-Emery liquid phase with \(c=1\) , divergent CDW and SC susceptibilities with the product \( K^{\vphantom{\dagger }}_{\rm CDW} K^{\vphantom{\dagger }}_{\rm SC}=1\)  and with the \(K_{\rm SC}\)  in the range \(0.5 < K_{\rm SC} \lesssim 2\) , and exponentially falling SDW correlations with \(\xi _{sdw} \sim 10\) . Where this occurs, it is typically also true that \(Q^{\vphantom{\dagger }}_{\rm CDW} = 4\pi x\) . The sketch in Fig. REF b is a caricature of the short-range order seen here. Recalling that periodic boundary conditions have been enforced around the cylinder, it is apparent that the SC order – which oscillates in sign between neighboring bonds around the cylinder – has a “true \(d\) -wave” form [7]}, [6]} in the sense that it changes sign under a \(C_4\)  rotation about the central axis of the cylinder. Such a state is likely to be a peculiarity of the 4-leg cylinder.
